Important items, like weapon parts and artifacts, are credited to both players no matter who picks them up. Players will receive their own ammo, health, stasis, and resource drops. Each player keeps their own valuable inventory at the end of a co-op session.
When you continue your individual game you'll be able to load your own uninterrupted campaign or load up your advanced save file. If you're just starting out you can jump into a buddy's Chapter 10 game, if you want. Story progress is saved for both characters. Carver's personal hell is invisible to Issac.